Repeating Ground Track Orbit - poliastro/poliastro GitHub Wiki

This is a list of all the algorithms taken into consideration to achieve a repeating ground track orbit.

Possible solutions:

David Eagle (MATLAB Central)

Here we have four possible solutions to design a repeating ground track orbit:

  1. Time to repeat ground track (nodal period) using Kozai orbit propagation.
  2. Time to repeat ground track using numerical integration. This solution uses a Runge-Kutta-Fehlberg
  3. Required mean semi-major axis using Wagner's algorithm. This one, it's quite similar to what it's explained in Vallado's book, Fundamentals of Astrodynamics and Applications 4th edition, pages (869-873). The PR #984 implements the algorithm detailed in Vallado.
  4. Required osculating semi-major axis using numerical integration. It's like the one mentioned in 2., but with the variant that the time to each ascending node is also determined by a Runge-Kutta-Fehlberg algorithm.

Curtis

  1. Curtis has a different approach (Pages 227-229), computing RA y Dec relative to the rotating Earth after a time interval Dt. It does not consider any perturbations.

Noreen 2003

  1. Noreen 2003 seems that the iterative algorithm taken from Vallado's doesn't seem to work with all the bodies. We can a find a slightly different solution to Mars.

Vtipil 2012

  1. Vtipil 2012 contains some mathematical background about the assumptions behind some other methods: Mortari "Flower constellations" (J2 only, any elliptic eccentricity), Aorpimai 2007 (J2, J4 and J2 ** 2, near-circular orbits) Vtipil 2010 and mentions a "simplified repeat groundtrack" (Collins 1977, J2).

All the examples to test the algorithm from PR #984 were obtained from:

  • Handbook of Satellite Orbits From Kepler to GPS by Michel Capderou.
  • Space Mission Analysis and Design, 3rd Edition, page 156. Specifically for the Earth case.

After some discussion, we agreed on implementing the Noreen algorithm for Sun-synchronous orbits to begin.

Assumptions:

  • Noreen paper
  • Circular orbits (no eccentricity)
  • Sun-synchronous orbit (link between semimajor axis and inclination)

Notes:

  • For Sun-synchronous orbits, if ecc = 0 we can fix a (and then i is given) or viceversa
  • For RGT, Sun-synchronous orbits, ecc = 0 comes from the assumptions, a comes from the period requirement, and this fixes i
  • Inputs: k, ndays, norbits (assumes ecc = 0)
  • Q = Number of periods (norbits), P_Q = Time to complete 1 of the Q periods (nodal period?), P_sid = Mars sidereal rotation, P_sol = Mars Solar rotation => Equation 2 gives P_Q in terms of the (1) sidereal and (2) solar rotation of the body and (3) the number of orbits
  • There's a relationship between P_K and a, and between n and a (third Kepler law)
  • Equation 10 from Noreen is similar to the second equation from Vallado pp. 870 (unnumbered): it relates the nodal period and the anomalistic period
P_Q = \frac{2 \pi}{n_t} = \frac{2 \pi}{n} \frac{n}{n_t} = &
P_K \frac{n}{n_t} = ...

but Noreen includes a factor cos(i) that is not present in Vallado.

  • Eqs. 6 to 8 are obtained with some assumptions with J2

Algorithm

  1. Inputs: Q, k, planet rotations, (ecc = 0)
  2. Q => P_Q from Eq. 2 assuming retrograde orbit (90º <= inc <= 180º)
  3. P_Q => n_t from Eq. 10
  4. Eqs. 15 and 18 relate (i, a). In Eq. 18, P_Omega = \frac{P_sol}{Q} and P_K = 2 \pi \sqrt{\frac{a^3}{k}}. But notice that Eq. 15 gives cos i directly, therefore it can be replaced inside Eq. 18 and this gives a relation between a and the input parameters of the problem (Q and the rotation of the attractor). Therefore, we solve for a.
  5. We use a in Eq. 15 to calculate i
  6. Check that the orbit is retrograde (90º <= inc <= 180º), and if not, ???
  • For the moment, raise an error